TUTORIAL REVIEW 2: Separation of Variables, Multipole Expansion, Polarization

INSTRUCTORS MANUAL: TUTORIAL REVIEW 2
Separation of Variables, Multipole Expansion, Polarization
Goals:
To revisit the topics covered in the previous 4 weeks of tutorials and cement concepts
prior to the exam
Timing:
This review may fall after tutorial 7, 8, or 9 depending on the semester and the midterm
exam schedule.
Reflections on this tutorial
Part 1
This problem took the students a lot of time considering this was the
second time they had done this problem. Students had trouble applying the boundary
conditions correctly and had difficulty figuring out which direction had waves as a
solution and which one had exponentials. Applying Fourier's trick in question(iii) was
still difficult for some students.
Part 2
Once again the Taylor series expansion proved to be the real challenge. A
lot of students remembered this problem from the previous weeks and knew how
everything was supposed to relate to the multi-pole expansion but just could not figure
out how to take the Taylor expansion. Similar problems came up again this week.
Students could not figure out what x0 and ε were supposed to be. Question(v) also
proved to be challenging once again. Some students chose by pattern matching and
needed instructor guidance to think through their choices.
Part 3
This section was put at the end as Polarization was covered just one week
earlier. Most students did not make it this far having taking a lot of time reviewing the
earlier material. One suggestion is to tell the students to do this tutorial in the order they
feel is best.
Tutorial Review 2, Week 11
© University of Colorado - Boulder
Instructor’s Manual
Contact: Steven.Pollock@Colorado.EDU
TUTORIAL REVIEW 2: Separation of Variables, Multipole Expansion, Polarization
 TUTORIAL REVIEW 2 
Separation of Variables; Multipole Expansion; Polarization
Please do this tutorial in whatever order you think would help
you most. You can start with Part 1, Part 2 or Part 3.
Part 1: Laplace’s Equation and Separation of Variables
Within a very long, rectangular, hollow pipe, there are no electric charges. The
walls of this pipe are kept at a known voltage (they are known because in a lab,
you can control them).
Three of the walls are grounded: V (x  0, y,z)  0 ; V (x, y  0,z)  0 ; V(x, y  b,z)  0
The fourth wall maintains a potential that varies with y: V(x  a, y,z)  Vo (y) which
will be specified later. 



V 0
yb
V  V0 (y)
V 0
y
V 0
x
xa
z
In order to find out the voltage inside the pipe, you will need to find a solution to
Laplace’s equation:
 2V  2V  2V
 V  2  2  2 0
x
y
z
2
i. Inspect the boundary conditions and choose which type of function will be used
for each variable (x, y, and z). For each choice, identify whether the separation
 negative, or zero.
constant is positive,
Tutorial Review 2, Week 11
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TUTORIAL REVIEW 2: Separation of Variables, Multipole Expansion, Polarization
ii. Apply boundary conditions #1-3 (the three grounded walls) and find the
general solution to Laplace’s equation which satisfies these 3 boundary conditions.
iii. The fourth wall is maintained at a constant voltage: V(x  a, y,z)  Vo . Does
your solution above satisfy this boundary condition? If not, will a sum of your
solutions satisfy the boundary condition? If so, give a formal expression for this

sum. If the sum involves coefficients for the individual solution terms, give a
formula for the coefficients. Any integrals should be in a form ready for
immediate integration (have limits and explicit variables and functions).
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TUTORIAL REVIEW 2: Separation of Variables, Multipole Expansion, Polarization
Part 2 – Potential from a Line of Charge
A uniform line charge density  extends from the origin to the point (0,0,-d).
Using the script-r technique from
earlier in the course, you can find an
expression for the potential V(z) for
points on the positive z-axis (z>0),
Remember:
V
1
4o

(r ')  d '
r
Your answer would be:

V (z) 

d
ln(1 ) . Don’t take
4o
z
time now to derive this solution (but be sure you can!)

i. If you were now asked to expand the potential, V(z), into a Taylor series for the
case z>>d, using the general formula below, what would you use for f(x), x0 and
? Recall the Taylor expansion about the point x  xo :
f ( xo   )  f ( x) x xo  f ' ( x) x xo  
f ' ' ( x) x xo
 2!
In this case the expansion is ln( 1   )   
2
2

 2  ...
3
3

ii. Use this to expand the potential V(z).
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TUTORIAL REVIEW 2: Separation of Variables, Multipole Expansion, Polarization
iii. For this charge distribution, calculate the monopole and dipole moments.




Recall that the dipole moment for a continuous distribution is p  r '  ( r ') d ' .
Identify which, if any, moments depend on the choice of origin.
iv. A potential can be expanded into the multipole form:
V
1
"monopole" "dipole" "quadrapole"


 ...) . Write the first two
4 o
r
r2
r3
(
terms of the multipole expansion using the monopole and dipole moments you just

calculated. Does the expansion agree with the Taylor series from part 2.i on the z-

axis? Recall that the numerator “dipole” = p  rˆ .
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TUTORIAL REVIEW 2: Separation of Variables, Multipole Expansion, Polarization
v. Without recalculating the potential using the script-r integral, but instead
thinking about the moments calculated in part 2.iii, how would the potential
change if the line charge distribution were shifted up by d/2, so it was centered at
the origin?
Part 3 – The Dielectric Sandwich
An infinite plane of charge with surface charge density f is between two infinite
slabs of neutral linear dielectric (of dielectric constant  and susceptibility ), as
shown. If the dielectric slabs were not present, the "bare" E-field, due only to the
plane of free charge, would have magnitude Eo = f / 20.
+++++++++++++++++++++++++++
++++
i. On the figure above make a simple sketch of any bound charge distributions you
expect (don’t perform any calculation yet).

ii. Again without calculation, sketch your expectation of the polarization P
everywhere, using little arrows.
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TUTORIAL REVIEW 2: Separation of Variables, Multipole Expansion, Polarization

iii. Does this arrangement of free charge and dielectric allow us to determine D
using Gauss’ law for the D field? Why or why not?


iv. Assuming that you agree that D can be determined using Gauss’ law, find D
everywhere in space.
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TUTORIAL REVIEW 2: Separation of Variables, Multipole Expansion, Polarization


v. Now find E and P everywhere in space.

vi. How does E “outside” the sandwich compare with the “bare” E field (no

dielectric slabs present)? Does your answer for P agree with your expectation
from part 3.ii?


vii. Calculate the bound charge distributions using  b  P  nˆ and b    P .
Do your distributions agree with your expectation from part 3.i?
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TUTORIAL REVIEW 2: Separation of Variables, Multipole Expansion, Polarization
viii) What do you expect for the boundary conditions on V and
V
at the
n
dielectric surfaces? How did you decide?


ix) Do your calculations for E and D give a V that satisfies these boundary
conditions?
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